Copyright © 2007 jsd

One Kind of Charge
John Denker

1  Overview

The most important things to know about charge are:

  1. There is only one kind of charge. The amount of charge can be positive, negative, or zero.
  2. Charge is conserved.
  3. Charge is the source of the electromagnetic field (as described by the Maxwell equations).
  4. In turn the electromagnetic field exerts a force each charge (as described by the Lorentz force law).

The first ideas is related to all the others. For starters, there is only one charge-conservation law. If there were two kinds of charge, we would need two conservation laws. Similarly, charge enters in only one way in the Maxwell equations and in the Lorentz force law.

Note the contrast:

Charge   Not Charge

If you are only talking about charge, then there is only one kind of charge, and you only need one number to keep track of the charge.   If you are talking about electrons and protons, then there are two kinds of particles, and you need two numbers to keep track of these two kinds of things.

The amount of charge might be positive or it might be negative, but it can’t be both at the same time. It’s only one number.   You can have some number of electrons and some number of protons at the same time.

Knowing the amount of charge will tell you the amount of charge.   Knowing the amount of charge will not tell you the number of electrons or the number of protons.

The idea of charge is completely abstract, not tied to any particular particles.   Electrons are not the only negatively-charged particles. Protons are not the only positively-charged particles.

Charge is charge. There is only one kind of charge.   There are lots of things we could talk about (e.g. momentum) that are not charge, and not even closely related to charge.

If you tried to define two kinds of electricity («vitreous electricity» and «resinous electricity»), you would find that the algebraic difference between the two is the only thing that matters in the charge-conservation law, the Maxwell equations, and the Lorentz force law. This algebraic difference is precisely the thing we call charge.

*   Contents

1  Overview
2  Comparing the Two Theories
3  Example: Atomic Physics
4  History and Logic
5  Conservation of Charge
6  Color Charge
7  Additional Discussion
8  Pedagogical Issues
9  References

2  Comparing the Two Theories

The fact is, there is only one kind of charge. We only need one variable – not two – to describe charge.

There is only one kind of electrical charge.

  However, in a great many textbooks and encyclopedias, one can find statements alleging that:

«There are two kinds of charge, namely positive charge and negative charge.»   

This expresses what is called the one-component model of electric charge, which is the recommended model.   This expresses the two-component model. We will have nothing good to say about it.

In reality, the charge can be represented as a point along a one-dimensional number line, as shown in figure 1. More positive charge is represented by points farther to the right on the number line, while more negative charge is represented by points farther to the left.   If we really did have two kinds of things – like apples and oranges – we would need two variables to describe the situation. For any object, the «charge» (or rather «charges») of the object would be represented by a point on a two-dimensional coordinate plane, as in figure 2.

Positive charge cancels negative charge and vice versa.   Apples do not cancel oranges nor vice versa.

number-line   apples-oranges
Figure 1: One-Dimensional Number Line   Figure 2: Two-Dimensional Number Plane
Charge   Apples and Oranges

The charge of the electron is different from the charge of the positron, but this is a difference in amount, not a difference in kind. The electron-charge is represented by a point on the number line, while the positron-charge is represented by another point on the same number line.  

The key point is that only need one variable to describe the charge of any given object. At any given time, this variable may be positive, negative, or zero, but it is still just one variable.   If there were two kinds of charge, we would need two variables to keep track of them.

Note: Some people use the word fluid, so they speak of a one-fluid model (as opposed to a two-fluid model). That’s essentially just another name for what we are calling the one-component model (as opposed to the two-component model). We prefer the latter name because charge isn’t exactly a fluid. It is an abstract quantity. It does, however, exhibit continuity of flow in close analogy to the flow of an indestructible fluid. Conservation of charge implies continuity of current. For more on this, see section 5 and reference 1.

Also note the following contrast:

In the real world, some quantities can be positive, negative, or zero. Consider for example elevation relative to sea level. If you look up the elevation of Furnace Springs, California, you will find it is a negative number.   To be sure, there are some quantitites that can never be negative, such as the number of apples in a basket. However, that does not change the fact that there are plenty of other quantities that can be positive, negative, or zero.

Some textbooks use the two-fluid language only in passing, using it as nothing more than a figure of speech, which isn’t so bad. However, others go out of their way to argue that the one-component model is wrong and the two-fluid model is right, which is appalling.

3  Example: Atomic Physics

Within the field of atomic physics, there is a sub-field where we deal with three kinds of elementary particles: protons, neutrons, and electrons. These things are different in kind. To keep track of them, we need three numbers.

Interestingly, the three numbers that are conventionally used are not the simple particle-counts, but rather the following:

  1. the baryon number (protons plus neutrons), denoted A
  2. the atomic number (protons), denoted Z
  3. the charge, denoted q

To fully specify an atom or atomic ion, it suffices to specify the three numbers (A, Z, q). For example:

If I tell you only that A=7 and Z=3, you have no idea what the charge is. You know it is lithium, but you have no idea whether it is ionized or not.

Conversely, if I tell you only that q=1, know the charge. The charge is 1 unit. You have no idea how many electrons, protons, or neutrons there are, but you know the charge. It could be Li+ or Na+ or Cu+ or lots of other things, but whatever it is, it bears one unit of charge.

To summarize: In this narrow sub-field, there are three kinds of elementary particles, but only one kind of charge. It takes three numbers to specify the particles, but it takes only one number to specify the charge.

More generally, there are dozens upon dozens of different kinds of particles, but still only one kind of electric charge. In physics, there are dozens of kinds of charged elementary particles but only one kind of electric charge; see figure 3. In chemistry, there are dozens upon dozens of charged atomic ions, not to mention molecular ions, but still only one kind of electric charge; see figure 4.

4  History and Logic

  In 1733, du Fay argued that there were two kinds of electricity, namely “vitreous” electricity and “resinous” electricity.

In 1747, William Watson argued in favor of a one-fluid model. Benjamin Franklin independently came to the same conclusion at the same time. Indeed, Franklin introduced the terms “positive”, “negative”, and “charge” for precisely this reason, to indicate a surplus or a deficit of the one type of electricity. See reference 2.   For many many decades after 1747, the two-fluid model remained viable. It wasn’t better than the one-fluid model, but it wasn’t significantly worse. It wasn’t provably wrong. It was a matter of opinion, on which reasonable persons could differ. In particular, we can restate the two-fluid model in modern terms: Let the number of protons represent the amount of “vitreous” electricity, and let the number of electrons represent the amount of “resinous” electricty. If you think electrons and protons are separately conserved, and if these are the only charge-carriers you know about, then you have a viable two-fluid theory.

In 1865, the Maxwell equations were published. These equations give an extraordinarily good description of classical electromagnetism. This is relevant because if you think the Maxwell equations are right, then you think that charge is conserved. It’s a strict mathematical corollary.   This doesn’t settle the argument, because if you think that “vitreous” and “resinous” electricity are separately conserved, then as a corollary, the difference of the two (i.e. the net charge) is also conserved, as in equation 1a.

In 1932, the positron was discovered, confirming a prediction made by Dirac in 1928. This invalidates the two-fluid theory. The number of electrons is not conserved. Protons are not the only carriers of “vitreous” electricity.  

An additional stake was driven through the heart of the two-fluid theory in 1964, when the subnuclear color charge was figured out. It is now very clear what an interaction looks like when it has to be described using more than one variable. See section 6. Note that color charge is completely independent of the ordinary electric charge. There is still only one kind of electric charge.  

So, for more than 80 years, it has been known that there are more than two types of charged particles, and the particles themselves are not separately conserved. The only electrical charge-like thing that is conserved is electrical charge itself.

Consider the following contrast:

  1. It is important to know what charge does. The most-important things we know about charge are :
    1. There is only one kind of charge.
    2. Charge is conserved.
    3. Charge is the source of the electromagnetic field (as described by the Maxwell equations).
    4. In turn the electromagnetic field exerts a force each charge (as described by the Lorentz force law).
  2. It is much less important, and more difficult, to figure out what charge «is».

5  Conservation of Charge

One indispensable property of charge is its role in the law of conservation of charge. This is the starting point and the linchpin for any understanding of what “charge” means. The law of conservation of charge states that the amount of charge in a given region cannot change except by flowing across the boundary of the region. This law can be stated with great precision and formality, and has been extensively checked and confirmed. It stands on its own as a fundamental law of nature ... and it can also be derived as a corollary of the Maxwell equations. For more on what we mean by conservation, see reference 1.

Suppose we start with a negatively-charged pion, which decays into a muon, which decays into an electron. There are also some neutrinos involved, but let’s not worry about them. They carry no charge.

Figure 3: Charge is Conserved. Particles Aren’t

The point here is that we recognize the charge as being the same charge, even though it is being carried by different particles.

Let’s see how this law applies to the following scenario: We have a closed, isolated, electrically-insulated container. Within the container there is a number Np of protons; as always, they carry one unit of positive charge apiece. There is also a number Ne of electrons; as always, they carry one unit of negative charge apiece. Finally there is a number Nn of free neutrons; they are electrically neutral. Note that neutrons that are not bound in a nucleus are unstable and undergo beta-decay with a half-life of just over 10 minutes. In this subsection, we temporarily pretend that other types of charge carriers (such as positrons) do not exist.

If the two-fluid model were correct, we would in general need two variables to keep track of the two kinds of charge. In the previous paragraph we did use two charge-related variables (Np and Ne) but we shall soon see that this was not necessary.

Let’s make a change of variables. For this system, we define

Q := Np − Ne    (for this system)      (1a)
R := Np + Ne       (1b)

The variable Q has a conventional name: it is called the charge. Equation 1 tells us how to calculate the charge for this system (whereas for a more complicated system a more complicated formula might be needed). The value of Q can be positive, negative, or zero. The law of conservation of charge states that Q is conserved.1

In contrast, the law of conservation of charge says nothing about R. In fact, R has nothing to do with charge, and is not even a conserved quantity. Every time a neutron decays, R increases by two, since the decay produces a new proton and a new electron. The charge Q is conserved during this process, as it must be for any process … but R is manifestly not conserved.

Let’s be clear: we only need one variable, Q, to tell us everything we need to know about charge. Any one-variable theory can always be dressed up to look like it has two (or more) variables, but there’s no point in doing so. It would be worse than useless.

Using the two variables Q and R is mathematically equivalent to using the two variables Np and Ne. If you start with Q, that is all you will ever need to keep track of charge; you don’t need R and you don’t need to keep track of Np and Ne separately in order to keep track of the amount of charge.

Of course R is meaningful; for example, an electrically-neutral plasma (Q=0, R≫0) has much greater electrical conductivity than an electrically-neutral vacuum (Q=0, R=0). (A parallel statement can be made concerning solid state physics: A compensated semiconductor is different from an intrinsic semiconductor.) That’s all fine; it simply tells you that there’s more to physics than just charge. Still, the fact remains that the single variable Q represents the charge, and you don’t need to know anything but Q to know the charge.

Tangential remark: It turns out that the quantity B := Np + Nn is conserved in this scenario. This is not a consequence of conservation of charge; it is a consequence of another conservation law, namely conservation of baryon number. Combining conservation of B with conservation of Q, we can infer that the quantity R + 2Nn is conserved … but R by itself is not conserved.

6  Color Charge

The nature of electrical charge can be understood with even greater clarity when contrasted with the subatomic color charge: There is only one kind of electrical charge, but there are three kinds of color charge.

Terminology: The term “charge” by itself is synonymous with plain old electrical charge. If you want to refer to color charge, you have to say “color charge”.

The symmetry of color charge is described by the group SU(3), while the symmetry of electrical charge is described by the group U(1). This allows us to say with great formality and precision that there are three kinds of color charge but only one kind of electrical charge.

For more on color charge, see reference 3, reference 4, and reference 5.

7  Additional Discussion

If we want to be scientific (as discussed in reference 6), due diligence requires that we examine the arguments in favor of the two-fluid model. We did some of this in section 4, but let’s go over it again.

One argument starts from the correct observation that in ordinary terrestrial matter, positive charge is predominantly embodied in protons, while negative charge is predominantly embodied in electrons. This however fails to prove that there are two kinds of charge. It fails for at least two reasons:

  1. The observation that protons are different from electrons is true as far as it goes, but it isn’t particularly relevant. It should be considered a statement about terrestrial matter, not as a statement about the fundamental properties of charge itself.

    Charge, like other fundamental physical quantities such as energy and momentum, is something of an abstraction. The fact that it is abstract doesn’t make it any less real. (See reference 7 for a discussion of abstract things and embodiments thereof.) We don’t say that momentum embodied in a piece of wood is in any fundamental way different from momentum embodied in a piece of plastic; the embodiment is different, but the momentum is fundamentally the same. By the same token, the charge embodied in electrons is fundamentally the same as the charge embodied in other particles. In particular, negatively-charged pions, muons, antiprotons, etc., do not “contain” electrons. They contain negative charge, but they do not contain electrons. Conversely, there are a lot of things we know about an electron, including mass, lepton number, charge, et cetera. So we see that the concept of charge and the concept of electron are very different. Charge is just one property of the electron, one property among many.

    This supports the crucial point of today’s discussion: If there were two kinds of charge (apples and oranges), you would need two numbers to describe the state of charge ... but this is never observed. You never need more than one number to describe the charge of any given object or region. To describe an electron you need more than one number, but that’s the answer to a different question. In particular, an electron is different from an antiproton, even though they have the same charge.

  2. It could be argued that electrons are observed to be less massive and more mobile than protons. Well, they are less massive, but they are only sometimes more mobile, so this is clearly not a fundamental distinction. Consider for example the electrical conductivity of a sample of one-molar nitric acid. The positive charge is carried predominantly by hydronium ions, while the negative charge is carried predominantly by nitrate ions, which are larger and less mobile!

    It’s a mistake to overemphasize the embodiments and/or the mobility of the embodiments. In a sample containing a mixture of five acids, there will be six different kinds of current. That alone should suffice to invalidate all the arguments in favor of a two-fluid model; if you’re going to allow more than one, logic requires you to allow an unlimited number.

    The smart way to proceed is to say that there is only one kind of charge, just as there is only one kind of momentum. The momentum is the same kind of momentum, no matter whether it is embodied in electrons, protons, wood, plastic, or whatever. By the same token, the charge is the same kind of charge, no matter whether it is embodied in electrons, protons, ions, subatomic particles, or whatever.

Charge is not matter, and matter is not charge. If you mean “charge”, you should say “charge”, while if you mean “charged particle”, you should say “charged particle”. There exist many kinds of charged particles, but only one kind of electrical charge.

Figure 4: Different Entities, Same Charge

The situation is shown in figure 4.

Within a given column, the entities on different rows have possibly different solubility, different lifetime, different mass, et cetera.   Within a given column, the entities have the same charge.

To keep track of many things, we need many numbers.   To keep track of charge, we need only one number.

8  Pedagogical Issues

It would be unwise to even mention the two-fluid model in an introductory course.

Such courses often explore the topic of electricity by means of simple macroscopic experiments. These may include rubbing a rubber rod on fur, peeling bits of cellophane tape off the desktop, and testing for the presence of charge using home-made electroscopes.

Macroscopic experiments of this sort do not provide any evidence whatsoever for preferring the two-fluid model to the one-component model. Even the fallacious arguments mentioned in section 7 are inapplicable in such a situation, because they involve microscopic issues that are not addressed by the introductory-level macroscopic experiments.

If we restrict attention to this sort of simple macroscopic experiments and nothing else, the two-fluid model and the one-component model are both equally-viable hypotheses. However, there is more to physics than simple experiments; there is also theory, and there are more advanced experiments. There are at least three ways to discriminate between the two-fluid model and the one-component model:

In an introductory class, there is no need to emphasize the one-component theory, nor to refute the two-fluid theory, nor even to mention the two-fluid theory (unless a student brings it up). The sensible approach is just to introduce the idea of “charge”, say that the amount of charge Q can be positive, zero, or negative, and proceed from there. This is the approach taken by some textbooks (e.g. reference 8). There is no need to make things more complicated than that. The simple and obvious answer is the fundamentally correct answer.

9  References

John Denker,
“Conservation as related to Continuity and Constancy”

Steven Weinberg,
The Discovery of Subatomic Particles Revised Edition

Sara Slater,
“What is the SU(3) quark model?”

Wikipedia article, “Special Unitary Group”

James Bottomley and John Baez,
“Why are there eight gluons and not nine?”

John Denker,
“Scientific Methods”

John Denker,
“The Black-Box Approach’

Sears, Zemansky, and Young,
College Physics, 6th edition
Addison-Wesley / Reading, MA (1985)

In this scenario, since the container is electrically insulated, Q is not just conserved, it is constant. See reference 1 for a discussion of the distinction between “conserved” and “constant”.

Copyright © 2007 jsd